IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
893
Modeling PCS Networks Under General Call
Holding Time and Cell Residence Time Distributions
Yuguang Fang, Student Member, IEEE, Imrich Chlamtac, Fellow, IEEE, and Yi-Bing Lin
Abstract—In a personal communication service (PCS) network,
the call completion probability and the effective call holding times
for both complete and incomplete calls are central parameters
in the network cost/performance evaluation. These quantities
will depend on the distributions of call holding times and cell
residence times. The classical assumptions made in the past that
call holding times and cell residence times are exponentially distributed are not appropriate for the emerging PCS networks. This
paper presents some systematic results on the probability of call
completion and the effective call holding time distributions for
complete and incomplete calls with general cell residence times
and call holding times distributed with various distributions such
as Gamma, Erlang, hyperexponential, hyper-Erlang, and other
staged distributions. These results provide a set of alternatives
for PCS network modeling, which can be chosen to accommodate
the measured data from PCS field trials. The application of these
results in billing rate planning is also discussed.
Index Terms—Billing rate planning, call blocking, call holding
time, call termination, cell residence, handoff, PCS.
I. INTRODUCTION
T
HE emerging personal communications services (PCS)
technologies have captured considerable attention in academic research as well as commercial deployment. A PCS
network can support a wide host of services when users are in
motion [2], [13], [20], [21] and can serve a large number of
customers by using spectrally efficient cellular systems [13],
[21]. In a PCS system, customers can make a phone call as in
wired telephony or make a connection to retrieve information
messages such as email or stock information or even make a
connection to surf the internet.
For billing and general performance tuning purposes, the
probability of a call completion and effective call holding
times need to be analyzed. To this end, the distributions of call
holding times and cell residence times need to be evaluated.
As a result of the new applications in the PCS networks
the classical assumptions on exponential call holding times
and cell residence times may not be appropriate for modeling
the new emerging integrated services in these systems. Field
Manuscript received December 1996; revised July 1997; approved by
IEEE/ACM TRANSACTIONS ON NETWORKING Editor S. Krishnan. The work
of Y. Fang and I. Chlamtac was supported in part by the U.S. Army Research
Office under Contract DAAH04-96-1-0308. The work of Y.-B. Lin was
supported in part by the National Science Council in Taiwan under Contract
NSC-87-2213-E-009-014.
Y. Fang and I. Chlamtac are with the Erik Jonsson School of Engineering
and Computer Science, The University of Texas at Dallas, Richardson, TX
75083-0688 USA.
Y.-B. Lin is with the Department of Computer Sciences and Information
Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 1063-6692(97)08936-X.
data for call holding times collected in office buildings and
residence areas in Taiwan show that the call holding times
cannot be modeled by exponential distribution, while Gamma
and lognormal distributions provide good (second-order) approximations to the experimental data. Not only does call
holding time distribution vary with the new applications, also,
the time a customer spends in a cell (the cell residence time)
will depend on the mobility of the customer, the geographic
situation, and the handoff scheme used, and therefore needs to
be modeled as a random variable of general distribution.
In order to facilitate our presentation of the modeling of
call holding times and cell residence times, we briefly review
the call connection process first. In a PCS system, the service
area is populated with base stations with the radio coverage of
each base station defining a cell, each with its assigned set of
customers. When a new call is originated by a customer in a
cell, one of the channels assigned to the base station is used for
the communication between the mobile portable and the base
station if a channel is available (a survey on channel allocation
schemes can be found in [10]). If all channels are in use while
a new call (or handoff call) is being attempted, the call will be
blocked and cleared from the system. If a call can be assigned
a channel, it will keep it until the call is completed or until the
mobile moves out of the cell. When the mobile moves into a
new cell while its call is active, a new channel needs to be
acquired in the new cell using a “handoff procedure.” During
the handoff, if no channel is available for the “old” call the
call will be forced to terminate before its completion [13].
The duration of the requested call connection is referred
to as the call holding time. When the call is connected, the
call may be completed after several successful handoffs, or
may be incomplete due to a failed handoff. We shall call the
duration of an incomplete call the effective call holding time of
an incomplete call and the duration of a call connection of a
complete call the effective call holding time of a complete call.
To evaluate the performance of a PCS network with appropriate rating programs such as flat-rate program [1], the
effective call holding times, as well as the probability of a call
completion, are required. From call completion probability,
we can also easily find the call dropping probability, another
important design parameter [4]. These quantities depend on
the distributions of call holding times and cell residence times
as well as the new call blocking probability and handoff
call blocking probability. Thus, the new call and handoff call
blocking probabilities can be regarded as inputs to the model.
Two main approaches to modeling call holding times and
cell residence times can be identified in the literature. One
1063–6692/97$10.00  1997 IEEE
894
uses the assumption of geographic cell (hexagonal) shapes
and the assumption of the constant mobiles’ speeds, random
distances, or uniformly distributed directions to determine the
distribution of cell residence times and call holding times [8],
[18], [26]. This approach faces a difficulty when applied to
existing cellular systems since, in reality, cell shapes are often
highly irregular, the speeds of mobiles or distances covered by
the mobiles are highly random (considering highly populated
area), and the directions of mobiles may vary in random
fashion. It becomes, therefore, very hard with this model to
characterize analytically the cell residence times using these
modeling assumptions.
The second approach treats the cell residence times and
call holding times directly, using information measured from
the PCS field trials. Distribution models such as exponential distribution, lognormal distribution have been used to
approximate the distributions of call holding times and cell
residence times using data from field tests (see [5], [16],
[17] and references therein). It is well known that exponential
distribution can be used for one-parameter approximation of
the measured data while Gamma distribution can be used for
two-parameter approximation [9]. It is also known [11] that
mixed Erlang distribution (hyper-Erlang distribution in this
paper) can be used to approximate any specific nonlattice
distributions. Hence, the application of these distributions to
model the call holding times and cell residence times in the
emerging PCS networks appears to be more practical when
field data are available.
In the traditional wired-line telephone models, the call holding times are usually assumed to be exponentially distributed,
which has been shown to be a reasonable approximation for
measured data. This assumption has been used in past PCS
network analysis for reasons of tractability [6], [8], [26], [27].
When call holding times are exponentially distributed, Lin et
al. [16], [17], Rappaport et al. [8], [22], [23], Yum and Yeung
[27], and Tekinay and Jabbari [25] studied the performance of
channel assignment strategies and obtained analytical results
for forced termination probability and new call blocking probability. Under the same assumption regarding call holding times
and general cell residence time distribution, Lin and Chlamtac
[15] obtained formulas for call completion probability and the
expected effective call holding times. When call holding times
are Erlang distributed and cell residence times have a general
nonlattice distributions, Fang, Chlamtac, and Lin [5] obtained
easily computable formulas for the call completion probability
and expected effective call holding times of a complete or an
incomplete call.
As emerging PCS networks are poised to provide various
new services they are expected to attract more users while
changing the users’ calling habits. Therefore, as pointed out
earlier, the call holding times are not expected to be distributed
with respect to the exponential (Erlang) distribution. More
general distributions for call holding times are, therefore,
needed to model these networks. In this paper, we propose
a general model that assumes that the cell residence times
have general nonlattice distribution and the call holding times
are distributed with general distributions such as Gamma,
hyper-exponential, and hyper-Erlang distributions. While the
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
advantages of such distributions are clear in that they reflect
the emerging services, applying these distributions to obtain
analytical results is a nontrivial task. This paper shows how
to accommodate general call holding time distributions in
previously proposed analytic models [5], [15]. The following
general assumptions will be used in this paper:
• the call arrivals form a Poisson process;
• the cell residence times are independent identically distributed (iid) with nonlattice distribution;
• the call holding times are independent identically distributed (iid) with nonlattice distribution.
Based on these assumptions, we obtain general formulas
for the call completion probability (hence, the call dropping
probability) and the distribution (its Laplace transforms) of the
effective call holding times of both the complete and incomplete calls from which expected effective call holding times
can be obtained. We derive computable formulas for the cases
when call holding times are distributed according to Gamma,
staged exponential or Erlang, hyperexponential, and hyperErlang distributions. Billing rate plans using the expected call
holding times for a complete call and an incomplete call are
also proposed and discussed briefly. The analysis of the call
completion probabilities and effective call holding times can
provide the necessary guideline for network performance evaluation tuning and importantly designing billing rate schemes
in the future PCS networks [1].
II. CALL COMPLETION PROBABILITY
In this section, we study the call completion probability.
Our previous work [5], [15] obtained formulas for the call
completion probability for a PCS network with a general
cell residence time distribution and Erlang (exponential) call
holding time distribution. Here, we give further results for
cases when the call holding times have other distributions.
New technique is developed for the case where the residue
theorem is not applicable.
We first consider the effective call holding time for an
incomplete call. Fig. 1 illustrates the timing diagram for the
call holding time,
is the time that the portable resides
at cell 1, and
is the residence time at cell
According to our assumptions,
are iid. Let
have nonlattice density function
with the mean
and
be the Laplace transform of
(we will use
to denote the Laplace transform following the tradition [12]).
Suppose that a call for the portable occurs when the portable
is in cell 1. Let
be the interval between the time instant
when the call arrives and when the portable moves out of cell
1. Let
and
be the density function and the Laplace
transform of
distribution, respectively. From the renewal
theory [12], is the residual life of the cell residence time of
the portable in cell 1, so we have
(1)
(2)
Let
and
be the effective holding time and
be its density function and Laplace transform.
FANG et al.: MODELING PCS NETWORKS
Fig. 1.
895
The timing diagram for a forced terminated call at k th handoff.
Since
Assume that the call holding times are Gamma distributed
with the following density function:
are independent, it is easy to derive
(3)
(6)
be the probability that a new call attempt is blocked
Let
(i.e., the call is never connected),
be the probability that a
call is completed (i.e., the call is connected and completed),
and
be the forced termination probability or the probability
that no radio channel is available when a handoff call arrives.
Then the call dropping probability or the call incompletion
probability (i.e., the call is connected but is eventually forced
to terminate)
is
which is given by
is the shape parameter and
is the scale
where
parameter. This density has the following Laplace transform:
(7)
Substituting (7) into (5), we obtain
(8)
(4)
is the density function of the call holding times.
where
In [5], we have obtained
In order to evaluate this integral, we need the following
lemma.
Lemma 1: Let
and
is analytic on
Re
Re
denotes the real part) with
as
in
denotes the
integral part of for any positive real number). Then, we have
(9)
where
when
Proof: The proof is given in the Appendix.
Now, we are ready to give an expression for
Let
(5)
has no poles in the right half complex plane,
Let
denote the set of poles of
in the right half complex plane (i.e.,
is
the set of poles of
in the left half plane). In [5], we have
obtained the call completion probability for the case when the
call holding times are Erlang distribution. Next, we study other
cases when the call holding times have other distributions.
(10)
Since
Note that
In fact,
is analytic on
and
on
satisfies all conditions in Lemma 1 and
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
Taking this into consideration in (8) and using Lemma 1, we
have
distribution) with parameters
distinct) whose Laplace transform is [12]
(which are
(14)
A random variable with this distribution is in fact the summation of exponentially distributed random variables with the
parameters
From this, we have the following.
Corollary 1: For a PCS network with -stage exponential
distribution of distinct positive parameters
the probability of a call completion is
from which we obtain the following theorem.
Theorem 1: For a PCS network with Gamma distributed
calling holding times, the probability of a call completion is
given by
(11)
(15)
Proof: From Theorem 2, we obtain
Res
Remark: When is a positive integer, the Gamma distribution becomes Erlang distribution. In this case
(12)
From (11) and (12), we obtain the same result obtained from
the residue theorem in [5].
If the call holding time has such a distribution that its
Laplace transform
has no branch points but has possible
isolated poles, then from the residue theorem we can obtain
[5].
Theorem 2: If the Laplace transform of the call holding
time distribution only has isolated poles in the left half of the
complex plane, then the probability of a call completion is
given by
Res
(13)
denotes the residue at the pole
In
where Res
particular, when
is rational function, then (13) is valid.
We have obtained the formula for the case that the call
holding times are Erlang distributed [5]. It is well known
[12] that the Erlang distribution can be obtained by a series
of independent identically distributed random variables. By
serial-parallel stages, a large number of general distributions
can be obtained from exponential distributed random variables
[12]. It is easy to observe that when
is a rational
function, then Theorem 2 can be easily applied to find
The distributions obtained by the method of stages belong to
this class. We will discuss some of the important cases next.
Let the call holding times be distributed according to
the -stage exponential distribution (the generalized Erlang
This completes the proof.
Assume now that the call holding times are distributed according to the -stage Erlang distribution with distinct parameters
and positive integers
,
which has the following Laplace transform:
(16)
This distribution can be obtained from the sum of independent Erlang distributed random variables with parameters
For this case, we have the following:
Corollary 2: For a PCS network with -stage Erlang distributed call holding times, we have
(17)
Equation (17) is specific for serial stages. For -stage parallel exponential distribution (the hyperexponential distribution)
FANG et al.: MODELING PCS NETWORKS
with the parameters
and
following Laplace transform [12]:
897
with the
call holding time of an incomplete call that is forced to
terminate is given by
(18)
Corollary 3: For a PCS network with hyperexponential
distributed call holding times, we have
(19)
More general cases can be obtained from serial-parallel
stages. One important case [12] is the hyper-Erlang distribution
with the following density function:
(20)
with the following Laplace transform:
(21)
This distribution is obtained from parallel Erlang distributed
random variables. It is shown [11] that the hyper-Erlang distributions can approximate any general (nonlattice) distribution.
Corollary 4: For a PCS network with hyper-Erlang distributed call holding times, we have
(22)
One general and interesting case obtained by the method of
stages is the distribution with Laplace transform
(23)
denotes the probability of a call to
where
be incomplete and
is computed in the previous section. In
the first equation, the term under the summation is the density
that the call is forced to terminate after handoffs.
We want to find the Laplace transform of
from which
the expected value can be easily obtained. As in [5], from
(17) we can obtain
(24)
Since
is a removable singular point [14] of the integrand of
(24). Thus, the poles of the integrand in the right half-complex
plane is those of
i.e.,
Let
(25)
Assume that the call holding times are Gamma distributed
as in (6) and (7). Taking (7) into (24), choosing to be greater
than the real part of and using Lemma 1, we have
A similar result for the probability of a call completion can be
derived, details are left to the reader. For the method of stages,
the interested reader is referred to [12]. For the computation
of
needed in the above, a recursive algorithm is
constructed in [5].
III. EXPECTED EFFECTIVE CALL HOLDING TIMES
In the preceding section we discussed the probability for a
call to complete. To fully characterize the performance of a
PCS network it is necessary to also know the expected elapsed
times for the complete and the incomplete calls (their socalled effective call holding times), respectively. In [5], we
have presented results for the effective call holding times of
complete and incomplete calls, in particular for the case when
the call holding times are Erlang distributed. This section
provides new results for other cases of interest.
We first consider the effective call holding time of an
incomplete call. As in [5], the density function for the effective
(26)
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
Using the well-known formula
(27)
Corollary 5: For a PCS network with call holding times
-stage exponentially distributed with Laplace transform as in
(14), then
we obtain the following theorem.
Theorem 3: For a PCS network with Gamma call holding
times, the Laplace transform of the density function of the
effective call holding times of an incomplete call is given by
(31)
(28)
and the expected effective call holding time of an incomplete
call is given by
The expected effective call holding time of an incomplete call
is given by
(32)
(29)
Here,
limit will be taken for
is an integer).
Proof: Since
(26) we have
when
and the
when
from
from which we complete the proof.
Remark: When
is an integer, the Gamma distribution
becomes the Erlang distribution, the above result will reduce
to our previous result [5].
If
does not have any branch points in the right
half complex plane, then the residue theorem can be used to
obtain certain computable results. In fact, from (24) we have
the following general result [5].
Theorem 4: If
only has isolated poles in the left half
of the complex plane, then
Res
Proof: By taking (14) into (30), we obtain
Res
This proves (31). Equation (32) can be obtained by
This completes the proof.
For -staged Erlang distributed call holding times, we have
the following result.
Corollary 6: For a PCS network with call holding times
distributed according to -stage Erlang distribution as in (16),
we have
(30)
In particular, when
is rational function, then (30) is
valid.
If the call holding times are -stage exponentially distributed, we have the following result.
(33)
FANG et al.: MODELING PCS NETWORKS
899
and the expected effective call holding time of an incomplete
call is given by
Corollary 8: For a PCS network with hyper-Erlang call
holding times, we have
(37)
and the expected effective call holding time of an incomplete
call is given by
(38)
(34)
Proof: Taking (21) into (30), we have
Proof: Taking (16) into (30), we have
Res
Res
This proves (33). Equation (34) can be proved by
For hyperexponentially distributed call holding times we
have the following.
Corollary 7: For a PCS network with hyperexponentially
distributed call holding times (as in (18)), we have
(35)
from which the corollary can be proved.
Next, we study the expected effective holding time for a
complete call. The timing diagram is shown in Fig. 2 in which
As before,
the call is completed when the portable is in cell
represents the effective call holding time for a complete call.
while if
If
Let
; then we have
For
and the expected effective call holding time of an incomplete
call is given by
(36)
(39)
For
(40)
Using a simple conditional probability argument, we can
of the effective call holding
obtain the density function
time of a complete call is given by
Proof: Taking (18) into (30), we obtain
(41)
where
Res
from which the proof can be easily completed.
For the hyper-Erlang distributed call holding times, we have
the following.
(42)
(43)
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
Fig. 2. The timing diagram for the effective call times of a complete call; the call completes after k handoffs.
corresponds to (39) and
corresponds to (40),
where
is the probability of nonblocking,
is the probability of no-forced termination. Equation (42) can
be derived from
, where
denotes the probability that the call is completed
in cell
and the effective call holding time is not exceeding
Rigorous derivation can be obtained following a similar
argument in [16].
The Laplace transforms
and
of
and
respectively, are [5]
Theorem 5: For a PCS network with Gamma distributed
call holding times, we have
(48)
and the expected effective call holding time of a complete call
is given by
(44)
(45)
(49)
(46)
Remark: When
is a positive integer, then Theorem 5
reduces to the result for the case when the call holding times
are Erlang distributed [5].
When
does not have any branch points and has only
finite isolated poles, an application of the residue theorem to
(44) and (45) leads to the following result [5].
Theorem 6: For a PCS network, the Laplace transform of
the density function of the effective call holding time of a
complete call is given by
Let
(47)
Assume that the call holding times are Gamma distributed
with the Laplace transform (7). Then, from Lemma 1 we have
the Laplace transform of the density function of the effective
call holding time of a complete call
Res
Res
from which we obtain the following theorem.
(50)
If the call holding times are -stage exponentially distributed, then we have the following.
Corollary 8: For a PCS network with the -stage exponential call holding times [see (14)], the Laplace transform of the
FANG et al.: MODELING PCS NETWORKS
901
and the expected effective call holding time of a complete call
is given by
effective call holding time of a complete call is given by
(56)
(51)
and the expected effective call holding time of a complete call
is given by
Finally, for the call holding times are hyper-Erlang distributed, we have the following.
Corollary 11: For a PCS network with call holding times
distributed according to hyper-Erlang distribution as in (20),
we have
(52)
(57)
Proof: This and the following few results can be proved
as those for the effective call holding times of an incomplete
call and are left to the readers.
If the call holding times are -stage Erlang distributed, then
we obtain the following result.
Corollary 9: For a PCS network with call holding times
distributed according to the -stage Erlang distribution (16),
then
(53)
and the expected effective call holding time of a complete call
is given by
(54)
If the call holding times are hyperexponentially distributed,
then the following result can be obtained.
Corollary 10: For a PCS network with call holding times
distributed according to hyperexponential distribution, then
(55)
and the expected effective call holding time of a complete call
is given by
(58)
IV. BILLING RATE PLANNING
In previous section, we derived analytical expressions for
expected call holding times for both complete and incomplete
calls. From these two quantities, we can easily compute
the expected call holding time, i.e., the expected time of
service usage for any call (either complete or incomplete).
The objective of this section, is to demonstrate the usefulness
of the effective call holding times not only for the performance
evaluation of the system, but also for effective service charging
planning.
Customers may have different calling habits in different
places at different times, creating the need for service providers
to determine the best rate plan. Based on customer’s usage,
service providers may try to adaptively change their charging
plans to provide the customer with an optimal rate. For a
service provider this may serve to gain customer satisfaction
and stay competitive.
As PCS networks are targeted to provide integrated services,
one can stipulate that it is not fair to charge the users for
the incomplete calls the same rate as for complete calls. For
example, in the FTP application, if the portable is forced to
terminate, the file needs to be retransferred. It may be more
reasonable for this loss to be shared by the customer and the
service provider. On the other hand, it may be considered
unfair for service providers to charge the same rate for short
and long incomplete calls. The possibility to differentiate
between service quality and charge accordingly is also in the
marketing interest of the PCS providers. Such differentiation
is not possible without solutions for the effective call holding
times unavailable in the past. Thus, cellular rating systems did
not differentiate between complete call and an incomplete calls
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
using the same rate for the air time, which can be determined
by the effective call holding time. Given our results above,
it becomes possible to distinguish between the air times for
complete calls and incomplete calls so that different rating
systems can be developed.
One may contend that it would be difficult to distinguish
the “true” forced termination from the false one. The user
may purposely terminate the connection, claiming a forced
termination. This issue does not exist in the current PCS
systems. A service provider only provides services within the
service area. In this area, the service provider is responsible for
the support of handoffs. The service provider is not responsible
for signal disconnection when the portable moves out of the
service area. Forced termination can be easily detected when
handoff fails due to the channels being assigned by the base
stations.
In the following, we briefly discuss a few possible service
charging (rating) systems for PCS network services.
A. Flat-Rate Planning
The simplest rate planning is the flat-rate plan, which applies
a flat rate to all calls (either complete or incomplete). This rate
is easy to implement and easy for customers to understand;
the determining factor for the charging rate is the expected
effective call holding time
where
and
Noticing
that
we have
In wireline
networks, where there is no forced termination
the flatrate planning is reasonable. Obviously, when the call dropping
probability is sufficiently small (this is the case when either
call traffic is light and mobility is comparably low or the
number of channels is sufficient to support all incoming calls),
flat rate planning is useful.
B. Partial Flat-Rate Planning
Due to the increase of the number of users and the mobility
of users, cell traffic will tend to increase. Hence, the effective
call holding times for incomplete calls may be significant.
In this case, one should consider using different rates for
complete calls and incomplete calls. Let
and
denote the
rates for an incomplete call and a complete call, respectively.
Then the average cost per call is given
(59)
From previous section, we know that the
Clearly,
expected effective call holding time
for an incomplete call
is longer than the expected effective call holding time
for
a complete call. Hence, from (59), we observe that slightly
increasing the rate
and decreasing the rate
does not
change the cost significantly. However, it is important to observe that customers are usually sensitive to call interruption,
and insensitive to the new call blocking. Thus, using discounts
for interrupted calls is a desirable policy, while these discounts
can be compensated by increasing the rate for complete calls,
to which the customers are less sensitive.
C. Nonlinear Charging Planning
In this case, the charges for complete calls and incomplete
calls are nonlinear functions of the effective holding times for
complete calls and incomplete calls. The cost per call may be
expressed as
(60)
and
are nonlinear functions. By choosing
where
these two functions, we can have different rate planning. It is
obvious if we choose
and
is a
convex function, then we have
which implies that the rating scheme using
and
separately can generate more revenues than a rate scheme which
uses the single parameter
The convexity of
implies
that we encourage medium length of calls, and discourage
very short calls and very long calls by applying a higher rate to
those calls (observing the shape of convex functions). Another
practical example is to choose the function
to be
while
may choose linear function. In this example, we
may choose the parameter
and
such that
These choices reflect the fact that the setup procedure is more expensive, while the longer calls are discouraged
in order to accommodate more users.
V. ILLUSTRATIVE EXAMPLES
This section presents some illustrative examples to show
how results obtained in this paper can be used to evaluate the
performance of PCS networks. We use following scenarios.
1) The cell residence times are iid according to the Gamma
distribution with parameter
and with different
values (change of mobility);
2) The call holding times are iid according to the following
distributions:
• exponential distribution with parameter
• Gamma distribution with parameters
;
• -stage exponential distribution with parameters
and
;
• hyper-exponential distribution with parameters
and
• hyper-Erlang distribution with parameters
and
;
;
.
is changing from 0 to 25.
3) The mobility
4)
and
typically values for current
cellular networks.
FANG et al.: MODELING PCS NETWORKS
Fig. 3. Call completion probability for different call holding time distributions.
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Fig. 5. Expected effective call holding time for an incomplete call.
Fig. 6. Expected effective call holding time for a complete call.
Fig. 4. Call completion probability; localized piece of Fig. 3.
In our examples, the considered distributions have the same
expected values (for exponential and Gamma distributions,
for -stage exponential distribution
their expectations are
for hyper-exponential and
its expectation is
hyper-Erlang distributions their expectations are
The choice of parameters above is to guarantee that
the average call holding times with different distributions are
all the same for comparisons, which are equal to
minutes, a value commonly used in the wired telephony trials
[13].
Figs. 3 and 4 show the call completion probability for
different call holding time distributions in the above scenarios
(the marking in Fig. 4 is used for curve reading). The following
can be observed.
• With low user mobility, the call holding time distributions
do not have significant impact on the call completion
probability (hence, the call dropping probability); with
high user mobility, the call holding time distributions do
have significant impact on the call completion probability.
• The call completion probability is always decreasing
as the mobility increases, which is consistent with our
intuition that the higher the mobility, the more the handoffs; hence, the higher the chance that the call will be
incomplete or the smaller the call completion probability.
These results are very important to the PCS network designers.
If the network is designed for low mobility, then the call
holding time distributions can be ignored. If high mobility
is expected in the PCS system, further analysis of the call
holding time distributions are required.
Figs. 5 and 6 show the expected effective call holding times
for a complete call and an incomplete call, respectively. The
vertical axis shows the normalized effective call holding times
the average call holding time when there is no
by the
new call blocking and no handoff call blocking (the ideal
case). Although the type of the call holding time distributions
does not significantly affect the call completion probability
for low user mobility, it does greatly affect the effective call
holding times for a complete call and an incomplete call. This
is why we should consider the effective call holding times in
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997
evaluating the performance of PCS networks. From these two
figures, we have the following observations.
• The expected effective call holding times (for either a
complete call or an incomplete call) are always decreasing
as mobility increases as expected intuitively.
• For the expected effective call holding time of a complete
call, the dependency on the type of call holding time
distributions increases as the mobility increases, however,
it is just the opposite for the expected effective call
holding time of an incomplete call.
• The expected effective call holding time of a complete
call is smaller than the absolute expected call holding
time (the ideal case when there is no blocking and no
forced termination), i.e.,
while the expected
effective call holding time of an incomplete call can be
larger than the absolute expected call holding time for
ideal case, for example,
for hyper-exponentially
distributed call holding times for low mobility. This
seems to be counter-intuitive. One can give the following
explanation, however: an incomplete call most likely goes
through many handoffs; hence, only “long” calls to be
dropped, therefore, the completion of “short” calls and
dropping of “long” calls display the above phenomenon.
The phenomenon that the expected effective call holding
time for an incomplete call tends to be longer than the
expected call holding time was not shown up in our
previous study [5]. In fact, for the case when call holding
times are exponentially distributed, we have shown [5]
analytically that the expected effective call holding times
for both a complete call and an incomplete call are shorter
than those of the ideal case.
contour
the function
theorem [14] we have
is analytic, hence, from the residue
from which we obtain
(61)
we have
Since
On the line
while on
From (61), by letting
VI. CONCLUSIONS
Previous performance studies of PCS channel allocation assumed that the call holding times are exponentially distributed.
While this assumption is justified for existing cellular systems,
future PCS systems will provide new types of services that will
affect the calling behavior of the users. Thus, a more general
distribution is desirable to model the call holding times. In this
paper, we use a general distribution to model the call holding
times and derive general formulas for the call completion
probability (hence, call dropping probability) and the expected
effective call holding times of both complete and incomplete
calls. For Gamma, (staged) Erlang, hyperexponential and
hyper-Erlang call holding time distributions, we obtain easyto-compute formulas to compute these quantities. These results
can be expected to become significant in evaluating and tuning
PCS network performance and help in designing new billing
rate programs for these networks.
APPENDIX
Proof of Lemma 1: If is not an integer, then
is
a branch point of the integrand [14]. Let us cut the complex
plane on the real axis from
right to the
as shown
in Fig. 7, where
and
In the domain enclosed by the
letting
from which the lemma is proved.
we obtain
FANG et al.: MODELING PCS NETWORKS
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Yuguang Fang (S’92–M’94–S’96) received the
B.S. and M.S. degrees in mathematics from Qufu
Normal University, Shandong, The People’s Republic of China, in 1984 and 1987, respectively, the
Ph.D. degree in Systems and Control Engineering
from Case Western Reserve University, Cleveland,
OH, in January 1994, and the Ph.D degree in
electrical and computer engineering from Boston
University, Boston, MA, in May 1997. Currently,
he is a Visiting Assistant Professor in Erik Josson
School of Engineering and Computer Science at the
University of Texas at Dallas.
From 1987 to 1988, he held research and teaching positions in both the
Mathematics Department and the Institute of Automation at Qufu Normal
University. From 1989 to 1993, he was a Teaching and Research Assistant
in the Department of Systems, Control, and Industrial Engineering at Case
Western Reserve University, where he became a Research Associate from
January 1994 to May 1994. He held a Postdoctoral Research Associate in
the Department of Electrical and Computer Engineering at Boston University
from 1994 to 1995. From 1995 to 1997, he was a Resaerch Assistant
in the Department of Electrical and Computer Engineering. His research
interests include wireless networks and mobile communications, personal
communications services (PCS), stochastic and adaptive systems, hybrid
systems in integrated communications and controls, robust stability and
control, nonlinear dynamical systems, and neural networks.
Imrich Chlamtac (M’86–SM’86–F’93/ACM F’96) received the Ph.D. degree
in computer science from the University of Minnesota (1979), and the B.Sc.
and M.Sc. degrees in mathematics awarded with the Highest Distinction.
He currently holds the Distinguished Chair in Telecommunications at the
University of Texas at Dallas. He is also President of Boston Communications
Networks, a company dealing with research and development of wireless
systems and high-speed networks. He is the author of more than 200 papers
in refereed journals and conferences, multiple book chapters, and coauthor of
Local Networks (Lexington Books, 1981).
Dr. Chlamtac is the founding Editor in Chief of the ACM-URSI-Baltzer
Wireless Networks (WINET) and the ACM-Baltzer Mobile Networking and
Nomadic Applications (NOMAD) journals, and served on the editorial board
of IEEE TRANSACTIONS ON COMMUNICATIONS and several other journals. He
served as the General Chair of several ACM and IEEE conferences and
workshops and is the founder of ACM/IEEE MobiCom. He is an IEEE Fellow
and an ACM Fellow, and in the past was a Fulbright Scholar and an IEEE,
Northern Telecom, and BNR Distinguished Lecturer.
906
Yi-Bing Lin received the B.S.E.E. degree from
National Cheng Kung University in 1983, and the
Ph.D. degree in computer science from the University of Washington in 1990.
From 1990 to 1995, he was with the Applied
Research Area at Bell Communications Research
(Bellcore), Morristown, NJ. In 1995, he was appointed as a Professor of Department of Computer
Science and Information Engineering (CSIE), National Chiao Tung University (NCTU). In 1996, he
was appointed as Deputy Director of Microelectronics and Information Systems Research Center, NCTU. Since 1997, he has been
elected as Chairman of CSIE, NCTU. His current research interests include
design and analysis of personal communications services network, mobile
computing, distributed simulation, and performance modeling.
Dr. Lin is a subject area editor of the Journal of Parallel and Distributed
Computing, an associate editor of the International Journal of Computer
Simulation, an associate editor of IEEE NETWORKS, an associate editor of
SIMULATION magazine, an area editor of ACM Mobile Computing and
Communication Review, a columnist of ACM Simulation Digest, a member
of the editorial boards of International Journal of Communications Systems,
ACM/Baltzer Wireless Networks, and Computer Simulation Modeling and
Analysis, Program Chair for the 8th Workshop on Distributed and Parallel
Simulation, General Chair for the 9th Workshop on Distributed and Parallel
Simulation. Program Chair for the 2nd International Mobile Computing
Conference, the publicity chair of ACM Sigmobile, Guest Editor for the
ACM/Baltzer MONET special issue on Personal Communications, and Guest
Editor for IEEE TRANSACTIONS ON COMPUTERS special issue on Mobile
Computing.
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 5, NO. 6, DECEMBER 1997